How would you curve the edges of a 3D shape?
Here's a 3D version of your Plane.



Add some subdivisions by pressing CTRL + "R" The more subdivisions you use the smoother the curve you are after will be.



Select two edges (You could also do one at a time). Turn on Proportional Editing and choose a "Falloff" type. I choose "Smooth" here.



Press "S" to scale and the Axis you want to scale on. I pressed "S" + "Z". Use the mouse wheel to affect more or less vertices.



Add a Subsurface Modifier and "Smooth Shading" if you want a smoother look. *You can use the same method with a plane.



EDIT You could also do this with a "Bezier Curve". Create your shape and change the curve to 2D to fill the shape.